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Properties

Label 2550.2.a
Level $2550$
Weight $2$
Character orbit 2550.a
Rep. character $\chi_{2550}(1,\cdot)$
Character field $\Q$
Dimension $52$
Newform subspaces $41$
Sturm bound $1080$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 41 \)
Sturm bound: \(1080\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2550))\).

Total New Old
Modular forms 564 52 512
Cusp forms 517 52 465
Eisenstein series 47 0 47

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)\(17\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(+\)\(-\)\(-\)$+$\(4\)
\(+\)\(-\)\(+\)\(+\)$-$\(4\)
\(+\)\(-\)\(+\)\(-\)$+$\(2\)
\(+\)\(-\)\(-\)\(+\)$+$\(3\)
\(+\)\(-\)\(-\)\(-\)$-$\(5\)
\(-\)\(+\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(+\)\(-\)$+$\(2\)
\(-\)\(+\)\(-\)\(+\)$+$\(4\)
\(-\)\(+\)\(-\)\(-\)$-$\(4\)
\(-\)\(-\)\(+\)\(+\)$+$\(2\)
\(-\)\(-\)\(+\)\(-\)$-$\(5\)
\(-\)\(-\)\(-\)\(+\)$-$\(5\)
\(-\)\(-\)\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(21\)
Minus space\(-\)\(31\)

Trace form

\( 52 q + 2 q^{3} + 52 q^{4} - 2 q^{6} + 52 q^{9} + O(q^{10}) \) \( 52 q + 2 q^{3} + 52 q^{4} - 2 q^{6} + 52 q^{9} - 8 q^{11} + 2 q^{12} - 8 q^{14} + 52 q^{16} + 8 q^{19} + 20 q^{21} - 8 q^{23} - 2 q^{24} + 2 q^{27} - 16 q^{29} + 24 q^{31} - 8 q^{33} - 4 q^{34} + 52 q^{36} - 8 q^{37} + 8 q^{38} + 28 q^{39} - 16 q^{41} - 4 q^{42} - 8 q^{43} - 8 q^{44} - 8 q^{46} - 16 q^{47} + 2 q^{48} + 76 q^{49} - 2 q^{51} - 16 q^{53} - 2 q^{54} - 8 q^{56} + 16 q^{57} + 16 q^{58} - 40 q^{59} + 8 q^{61} - 24 q^{62} + 52 q^{64} + 8 q^{66} + 32 q^{67} + 4 q^{69} + 8 q^{71} + 32 q^{73} + 16 q^{74} + 8 q^{76} + 4 q^{78} + 8 q^{79} + 52 q^{81} + 24 q^{82} + 24 q^{83} + 20 q^{84} + 8 q^{87} + 48 q^{91} - 8 q^{92} + 36 q^{93} - 2 q^{96} + 40 q^{98} - 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2550))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 17
2550.2.a.a 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-3\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-3q^{7}-q^{8}+\cdots\)
2550.2.a.b 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-2q^{7}-q^{8}+\cdots\)
2550.2.a.c 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
2550.2.a.d 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
2550.2.a.e 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
2550.2.a.f 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(-1\) \(0\) \(3\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+3q^{7}-q^{8}+\cdots\)
2550.2.a.g 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(-1\) \(0\) \(5\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+5q^{7}-q^{8}+\cdots\)
2550.2.a.h 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(1\) \(0\) \(-3\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-3q^{7}-q^{8}+\cdots\)
2550.2.a.i 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(1\) \(0\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-2q^{7}-q^{8}+\cdots\)
2550.2.a.j 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
2550.2.a.k 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
2550.2.a.l 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
2550.2.a.m 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(1\) \(0\) \(1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+q^{7}-q^{8}+\cdots\)
2550.2.a.n 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(1\) \(0\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+4q^{7}-q^{8}+\cdots\)
2550.2.a.o 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(1\) \(0\) \(4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+4q^{7}-q^{8}+\cdots\)
2550.2.a.p 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(1\) \(0\) \(4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+4q^{7}-q^{8}+\cdots\)
2550.2.a.q 2550.a 1.a $1$ $20.362$ \(\Q\) None \(-1\) \(1\) \(0\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+4q^{7}-q^{8}+\cdots\)
2550.2.a.r 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(-1\) \(0\) \(-4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots\)
2550.2.a.s 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(-1\) \(0\) \(-4\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots\)
2550.2.a.t 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(-1\) \(0\) \(-4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots\)
2550.2.a.u 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
2550.2.a.v 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(-1\) \(0\) \(-1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-q^{7}+q^{8}+\cdots\)
2550.2.a.w 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
2550.2.a.x 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(-1\) \(0\) \(1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
2550.2.a.y 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(-1\) \(0\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+2q^{7}+q^{8}+\cdots\)
2550.2.a.z 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(-1\) \(0\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+3q^{7}+q^{8}+\cdots\)
2550.2.a.ba 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(1\) \(0\) \(-5\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-5q^{7}+q^{8}+\cdots\)
2550.2.a.bb 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(1\) \(0\) \(-3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-3q^{7}+q^{8}+\cdots\)
2550.2.a.bc 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(1\) \(0\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)
2550.2.a.bd 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(1\) \(0\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{8}+q^{9}+\cdots\)
2550.2.a.be 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(1\) \(0\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+2q^{7}+q^{8}+\cdots\)
2550.2.a.bf 2550.a 1.a $1$ $20.362$ \(\Q\) None \(1\) \(1\) \(0\) \(3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+3q^{7}+q^{8}+\cdots\)
2550.2.a.bg 2550.a 1.a $2$ $20.362$ \(\Q(\sqrt{33}) \) None \(-2\) \(-2\) \(0\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-\beta q^{7}-q^{8}+\cdots\)
2550.2.a.bh 2550.a 1.a $2$ $20.362$ \(\Q(\sqrt{5}) \) None \(-2\) \(2\) \(0\) \(-2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+(-1-\beta )q^{7}+\cdots\)
2550.2.a.bi 2550.a 1.a $2$ $20.362$ \(\Q(\sqrt{6}) \) None \(-2\) \(2\) \(0\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+\beta q^{7}-q^{8}+\cdots\)
2550.2.a.bj 2550.a 1.a $2$ $20.362$ \(\Q(\sqrt{6}) \) None \(2\) \(-2\) \(0\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+\beta q^{7}+q^{8}+\cdots\)
2550.2.a.bk 2550.a 1.a $2$ $20.362$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(0\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+(1+\beta )q^{7}+\cdots\)
2550.2.a.bl 2550.a 1.a $2$ $20.362$ \(\Q(\sqrt{6}) \) None \(2\) \(2\) \(0\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+\beta q^{7}+q^{8}+\cdots\)
2550.2.a.bm 2550.a 1.a $2$ $20.362$ \(\Q(\sqrt{33}) \) None \(2\) \(2\) \(0\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+\beta q^{7}+q^{8}+\cdots\)
2550.2.a.bn 2550.a 1.a $3$ $20.362$ 3.3.568.1 None \(-3\) \(-3\) \(0\) \(-6\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+(-2-\beta _{1}+\cdots)q^{7}+\cdots\)
2550.2.a.bo 2550.a 1.a $3$ $20.362$ 3.3.568.1 None \(3\) \(3\) \(0\) \(6\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+(2+\beta _{1})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2550))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2550)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(255))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(425))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(510))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(850))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1275))\)\(^{\oplus 2}\)